Simulation Setup¶
Below can be seen two images that setup the problem we are looking at. The diagram of the model can be seen with the excited base moving up the Y axis and inverted mass connected with a rod free to move in the X-Y plane. The mass is only influenced by the force of gravity and restricted by the connection to the exicted base. The angle $\theta$ is defined as positive from the vertical Y-axis moving clockwise in the diagram.
The bond graph below shows the physical model of the system and how each part is interacting such that we can derive equations to solve for the motion of the inverted mass.
Equations used for simulations¶
Velocity equations for X, Y, and $\theta$ directions
$$v_y = \dot{y} - l \dot{\theta} \sin(\theta) $$$$v_x = l \dot{\theta} \cos(\theta) $$$$ \dot{\theta} = \frac{p_x}{l\cos(\theta)\ m} $$Momentum equations
$$p_x = m l \dot{\theta} \cos(\theta) $$$$p_y = m\left[\dot{y} - \frac{\sin(\theta)\ p_x}{\cos(\theta)\, m}\right] $$Derivatives of momentum equations $$\dot{p_y} = m\left[\ddot{y} - \frac{\sin(\theta)\ \dot{p_x}}{\cos(\theta)\ m} - \frac{p_x\ \dot{\theta}}{m\ \cos^2{\theta}} \right] $$ $$\dot{p_x} = m g \sin(\theta)\cos(\theta) + m \ddot{y}\sin(\theta)\cos(\theta) - \frac{\sin(\theta)}{\cos(\theta)} \dot{\theta} p_x$$
